Dynamic resonance

The equations of motion for the energy levels derived in Section 4.1.2 can easily be generalized to the case of motion in the complex plane if 

Note that the scalar product (10.5.13) does not involve complex conjugation and is therefore not positive definite. The final set of differential equations describing the motion of the helium resonances in the complex energy plane is identical with the set (4.1.57) with the only difference that the quantities appearing in (4.1.57) are now complex. 

The resonances at e = 0.5 are shown as the full circles in Fig. 10.14. In order to determine their local velocities at e = 0.5, we also computed the location of these resonances in the complex plane at e = 0.51. The resulting locations at e = 0.51 are shown as the open circles in Fig. 10.14. As expected the resonances move in the complex plane as a function of e. Moreover, the movement is not only in one direction. Especially close to the real energy axis, the resonances move in different directions. This is already close to what we expect from a gas of particles. A finite difference approximation to the local velocity of the resonances at e = 0.5 is computed according to 

This shows that the resonance velocities contain a collective component. The data peak at = 2.15 x 10. This corresponds to a collective drift whose origin and magnitude can be understood qualitatively. Using the independent particle model for a rough estimate of the energy levels we have 

The mean n value, n, of all the resonances considered is fi 15. Therefore, w 2.2 X 10 , which agrees well with the observed collective drift. 

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Kellman, M. E. (1985), Algebraic Resonance Dynamics of the Normal/Local Transition from Experimental Spectra of ABA Triatomics, J. Chem. Phys. 83,3843. 

Martens, C. C., and Ezra, G. S. (1987), Classical, Quantum Mechanical, and Semiclassi-cal Representations of Resonant Dynamics A Unified Treatment, J. Chem. Phys. 87, 284. 

We first consider the AN regime of a two-level system coupled to a thermal bath. We will use off-resonant dynamic modulations, resulting in AC-Stark shifts (Figure 4.5(a)). The Hamiltonians then assume the following form ... 

Obviously, much information on the resonance dynamics should normally be gained by close examination of the structure of the wavefunction in the internal region of the configuration space. However, it is often possible also to infer either the dominant roles or the secondary roles of the interaction potentials in resonance processes without going into details of the wave-function. The inspection of the potentials can often lead to transparent visual understanding of the essential dynamics. 

Edwards, T. E., and Sigurdsson, S. T. (2002). Electron paramagnetic resonance dynamic signatures of TAR RNA—Small molecule complexes provide insight into RNA structure and recognition. Biochemistry 41, 14843-14847. 

The methods of level dynamics can be extended to resonance dynamics in case the energy levels En acquire a width in the presence of a continuum. Resonance dynamics of the one-dimensional helium atom is discussed in Section 10.5.2. 

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These five cases are excellent examples of the additional challenges that can arise In the study of shape resonance phenomena. They should not diminish the simplicity and power of the fundamental shape resonance dynamics but, rather, should show how the fundamental framework showcases more complicated (and Interesting) photolonlzatlon dynamics which. In turn, require a more sophisticated framework for full understanding. 

Near the resonant energy E = 1.806 eV, the channels which participate most In the resonance dynamics are clearly "0 of the H2 reactants and Vp>2 of the HF products. 

Sessin and Ferraz-Mello, 1984). This integral of the resonant dynamics means that ai and a2 vary in anti-phase. When one of the semi-axis increases, the other necessarily decreases. 

Figure 8.5 Coherence resonance. Dynamics of an excitable system perturbed by stochastic fluctuations for different values of the noise amplitude increasing from top to bottom. At intermediate noise levels, nearly periodic spikes occur (from Pikovsky and Kurths (1997)).

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